**Solution:**

Let's assume that the dealer mixes tea A (which costs 9 per kg) and tea B (which costs 12.6 per kg) in the ratio of 'a' and 'b' respectively.

To find the ratio of 'a' and 'b', we need to consider the cost and selling price of the mixture.

**1. Cost of the Mixture:**
The cost of the mixture is the sum of the cost of tea A and tea B in the given ratio.

Cost of tea A = 9a (as tea A costs 9 per kg)
Cost of tea B = 12.6b (as tea B costs 12.6 per kg)

So, the total cost of the mixture = 9a + 12.6b

**2. Selling Price of the Mixture:**
The selling price of the mixture is given as 13.5 per kg. Since the dealer wants to realize a profit of 20%, the cost price of the mixture can be calculated as follows:

Cost price of the mixture = Selling price / (1 + Profit%)

Given that the selling price is 13.5 per kg and the profit is 20%, the cost price of the mixture = 13.5 / (1 + 0.2) = 11.25 per kg

**3. Equating the Cost Price and Cost of the Mixture:**
Since the cost price of the mixture should be equal to the total cost of the mixture, we can set up the equation as follows:

11.25 = 9a + 12.6b

**4. Finding the Ratio of a:b:**
To find the ratio of 'a' and 'b', we can simplify the equation:

11.25 = 9a + 12.6b
1125 = 900a + 1260b (multiplying both sides by 100 to remove decimals)

Dividing the equation by 225, we get:

5 = 4a + 5.6b

Now, let's assume that 'x' is a common factor of 'a' and 'b'. So, we can write 'a' as 5x and 'b' as 7x.

Substituting these values into the equation, we get:

5 = 4(5x) + 5.6(7x)
5 = 20x + 39.2x
5 = 59.2x
x = 5 / 59.2
x ≈ 0.084

Therefore, the ratio of a:b = 5x : 7x = 5(0.084) : 7(0.084) = 0.42 : 0.588

Simplifying further, we get the ratio of a:b ≈ 2 : 3 (approximately)

Hence, the dealer should mix tea A and tea B in the ratio of 2:3 to realize a profit of 20% by selling the mixture at 13.5 per kg.